Internat. J. Matil. & Math. Sci.
VOL. i4 NO.
(1991) 27-44
27
SOME PROPERTIES OF THE FUNCTIONAL EQUATION
(x)
|
i’(x) +
g(x,y,(y)) dy
0
LI. G. CHAMBERS
School of Mathematics
University College of North Wales
Bangor, Gwynedd, Wales LLS? IUT.
(Received February 7, 1989 and in revised form April 9, 1990)
ABSTRACT.
A discussion is given of some of the properties of the functional
Volterra Integral equation
Ax
O(x)
I0
fCx) +
gCx, y,O(y)) dy
Sufficient conditions are
and of the corresponding multidimensional equation.
given for the uniqueness of the solution,
for the construction of the solution,
and an iterational process is provided
together with error estimates.
In addition
The results obtained are illustrated by
bounds are provided on the solution.
means of the pantograph equation.
KEY WORDS AND PHRASES.
Functional Integral Equation.
1980 AMS SUBJECT CLASSIFICATION CODES.
i.
45D05, 45GI0, 45LI0, 3K05.
INTRODUCTION.
A certain amount of attention has been paid
equations
[I]
but
it
appears
very
that
little
to
functional
attention
has
differential
been
paid
to
functional Volterra Equations such as
Ax
(x)
f(x) +
I0
g(x,y,(y)) dy
0 <
The first remark which may be made is that if
uniqueness.
(1.1)
0 < x
A >
there may not be
For consider the equation
x
Differentiation
shows
that
this
integral
equation
is
equivalent
to
the
differential equation problem
@’(x)
a
@(Ax)
(l.2b)
with
(o)
(1.2c)
LL. G. CHAMBERS
28
and it is well known that the problem defined by (1.2b) and (1.2c) does not have a
unique solution [2].
A
In this paper, only values of
0 < A K
such that
will be considered.
Sufficient conditions will be obtained for the solution of the integral equation
a number of bounds will be found for the solutions of the
(I.I) to be unique,
will be given for the
equation and an iterational process, with error analysis,
These results will be valid provided
construction of the solution.
and
The relevant conditions will be
obey certain conditions as follows.
g(x,y,z)
f(x)
mentioned before each piece of analysis.
A)
Ig(x,y,z 1)
Iz
g(x,Y, Z2)l g P(x) Q(y)
(1.3)
Q(y) > 0
P(x)
z21
In part of the analysis
P(x)
y
Q(x)
px
p > 0
(1.4)
2:0
2:0
=
(In some places an alternative condition on
+
#
> 0
+
is used.)
The equation (I.I) may be rewritten as
(x)
f(x) +
r
Ax
g(x,y,o) dy +
0
f (x) +
[
r
Az
[g(x,y,(y))
g(x,y,o)] dy
(1.5a)
0
AX
(l.5b)
g (x,y,#(y)) dy
0
Clearly it follows from the inequality (1.3) that
Ik(x)l <
c)
(1.5c)
P(x) Q(y) IO(y)l.
Ig (x,y,O(y))l
B)
MA’x S
8 2:0
’ 2:0
(1.6a)
M > 0
where
Ax
r
k(x)
(1.65)
g(x,y,f(y)) dy
0
If
the theory is that of the usual Volterra equation which is
A
well known.
2.
UNIQUENESS
It may be shown that,
suppose that
A(x)
CB(x)
if condition
A
For
holds there is uniqueness.
are two possible solutions
(y)
g(x,y
(x,y,
(y)
(y))
g(x,y,,B(y))]
(2.1)
dy
0
<
x Ig(x,Y @A
dy
0
P(x)
Q(y)]OA(y)
(y)]
(2.2)
dy
0
By using the processes similar to
Gronwall’s inequality [3], it follows that
thus uniqueness.
those
involved
leA(X) B(X)
in
the
proof
of
vanishes and there is
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
3.
29
BOUNDS ON THE SOLUTION
It is possible, using Gronwall’s inequaliy to obtain functions which bound
the solution of equation (].I).
a)
Suppose that condition
A
holds.
The equation (I.I) can be rewritten as
Ax
(x)- f(x)=
Ax
r
g(x,y,f(y)) dy +
0
IJO
{g(x,y,(y))- g(x,y,f(y))} dy
Ax
/
k(x) +
{3.11
g{x,y,f(y)l} dy
{g(x,y,(y))
0
It follows that
Ax
IC)
fCx)l
IkCx)l
IgCx, y,Cyi)
+
-gCx, y, fCy))
dy
0
and using condition
lCx)
A
it follows that
IkCx)l
fCx)l
[
+
Ax
Q(y)lCy)
PCx)
fCy)l
(3.2)
dy
0
Let
P(x) O(x)
IO(x) f(x)
Ik(x)J P(x) h(x)
(3.3b) and (2.3b),
Using the relations (3.3a),
(3.3a)
(3.3b)
the inequality
(3.2) may be
rewritten as
Ax
(3.4a)
U(y) @(y) dy
whence
x
(3.4b)
U(y) O(y) dy
0
The inequality (3.4b) is in a form suitable for the application of Gronwall’s
O(x) K h(x) +
U(x)
Multiplying by
inequality.
and writing the result as a first order
differential relation for
x
/
@Cx)
(3.4c)
UCy) @(y) dy
0
it follows that
r
r
u(y),(y)dy
0
U(z) dz
h(y) Y(y)
dy
(3.51
dy
(3.6)
y
0
It follows from the inequalities (3.4a) and (3.5) that
O(x)
h(x) +
[
U(z) dz
h(y) U(y)
0
whence there follows the bounding inequality
I(x)
f(x)
Ik(x)
/
[
P(x)
I(y)l
Q(Y)
PCz) Q(z) dz
exp
dy.
y
0
(3.7a)
In the special case
P(x)
p
Q(y)
right hand side of (3.7a) becomes
and
I (x)l
s
differentiable the
LL. G. CHAMBERS
30
IkCx) -IkCx}
IkCY)le p (Ax-y)
d
IkCo)lJx+
+
(3 7b)
dy.
0
b)
B
Suppose that condition
Then, using the relations Cl.5b) and
holds.
.
(1.5c), it follows that
Ax
/;
ICxl I Cxl
It will be noted that if
@(x)
Icl
CxC
c.
is zero the inequality (3.8)
f (x)
implies that
vanishes.
By exactly the same process as
the
(3.7a)
inequality
was
deduced
from
the
inequality (3.2), it follows that
]#Cx)]
K
]f
(x)
;
+ PCx)
]f (yl]
PCz) QCz) dz
QCy)
C3.9a)
dy
y
0
and in the special case mentioned above, the right hand side of this becomes
,
If Cx)l- If Cx)l
If CO)I ePAX
+
A simplification occurs Lf
#(x)
K
If
(x)
P(x)
f
+
Ax
d
o @
If (y)lepC’x-y)
dy
C3.9b)
QCz) dz
dy
C3. lOa)
when
If (y)
0
+
QCy)
y
which can be rewritten as
+
o
if
If
(x)
-
If Cx)l- If Cx)l
ICx)l
If
+
If Co)I
QCz) dz
exp
0
Q(z) dz
(y) exp
dy
(3. lOb)
C
are satisfied, the
y
is dtfferentiable.
TIT IVE PROCF.SS
and
A B
4.
CONSTRUCT ON OF SOLUT IONS BY AN
a)
It wii1 now be shown that, if conditions
sequence of functions
#n(X)
defined by
Ax
#n+ICx)
fox) +
0
gCx, Y, On(y)) dy
@0(x)
converges to the solution
#(x)
(4.1a)
(4. Ib)
f(x)
of equation (I.I).
this sequence to obtain a further bound for
start with an arbitrary
n > 0
l#(x)
It is also possible to use
(In fact it is possible to
#o(X)
in which case there wil be slightly different
n(X)
]#(X)
results.
Let
XnCX)
may be regarded as the error
@n(X)l
involved
(4.2)
in
stopping at
the
nth
iteration.
Then
,Ax
n+iCx)
J
{gCx, y,(y))
0
g(x,Y,@n(y))}
dy
(4.3a)
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
31
x
px
y n(y)
=
(4.3b)
dy
Now
Suppose that the range of interest is
0 K x K c
Then it follows that,
using the inequality (3.7a),
Zo(X)
where
over
k
0
k
max
x
max
+ P
Pmax
max
[0
Q(Y)
PCz)
dz
dy
(4.4a)
y
are respectively the maximum values of
Ik(x)l
and
P(x)
c
C(c)
(4.4b)
say
Look for a solution of the recurrence inequality (4.3b) of the form
s
Xn(X)
C
n
A
n
t
x
n
(4.5)
Substitution of the inequality (4.5) in the inequality (4.3b)
Ax
n+l (x)
px
pC
A
y Cn A n y n dy
f
0
s
n
s
pC
A
n
n
n
x
x
y
(4.6a)
tn+ dy
t +6+1
tAX) n
+
t +
n
x
Eives
t
s
(4.6b)
The form (4.5) will be preserved if
pC
Cn+l
tn+
Sn+
t
n
n
+
(4.7a)
+
tn + = +
Sn + tn +
+
(4.7b)
+
(4.7c)
Comparison with the relation (4.4b) gives
C
O
C
(4.8a)
t
o
0
(4.8b)
s
o
0
(4.8c)
Solution of the recurrence relations (4.7) with the initial conditions (4.8)
gives the following results
C
pnc
n
(4.9a)
n-1
1-1"
.--0
+"
+
’1
32
LL. G. CHAMBERS
t
s
n(n
n
I)
2
n
n( +
+ 1)
( +
(4.9b)
n( + I)
+ I) +
C4 9c)
and thus the error in stopping at the nth approximation has been defined by (4.5)
and (4.9).
x g c
These results hold for
In particular they hold for
x
c
and so it follows that
s
n
), < p A
n-1
XnCC
t
n
c
1-r
n
C(c)
+"
(4.9d)
’}
+
,’-0
c
however is arbitrary and can be replaced by
p
n(X
s
n
A
t
n
x
n
x
and so
C(x)
(4.9e)
n-I
1-1"
.--0
C(x)
where
that if
A K
It can easily be verified
is defined by the relations (4.4).
and
=
+
> 0
+
the sequence
Xn(X)
converges to zero.
The sequence functions can also be used to determine a bound for
l#(x)
Consider the sequence
n
n+lCX)
lCX)
[ @mCX)
m=l
(4.1On)
#m(X}(4. lOb)
(4. lOb)
g(x,y,f(y)) dy
(4.10c)
+
where
@m (x)
#m+l(X)
and
f(x) + k{x)
(4.10d)
Then
n
If the series
n
converges, it dominates the series
n
ml @m(X)
which must also converge.
Therefore the sequence
#n+l(X)
converges and it
follows from the relation (4.1a) that the limit is the solution
(I.I).
l#(x)
(x) of equation
It follows from the inequality (4.10d) that there is a further bound to
namely
Following the analysis of equations (4.3) it can easily be shown that
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
33
x
(4.12a)
@o(X)
#l(X)
f(x)
#l(X}
#o(X)
Ax
I0
g(x,y,f(y)) dy
k(x)
(4.12b)
and so
l@o(X)
S
M AT x
K
(4.12c)
In exactly the same way as previously consider the inequality sequence
u
v
n
n
l@n(X)l Mn A x
(4.13)
Using the relation (4.12a) in the same way as the relation (4.6a) was used
pMn
Mn+l
Vn+l
v
+6+
vn +
+
Un
Un+
+v
(4. 14a)
(4.14b)
+
(4.14c)
n +6+1.
The initial relations are now
M0 =M
(4. 15a)
Vo=S
(4. 15b)
Uo=
(4. 15c)
Thus
n
p
M
n
M
(4. 16a)
n-1
=0
v
u
n
n(n
I)
n
(= +
2
(4.16b)
+ 1) +
n(= +
+ I) + n( +
(4. 16c)
+ I) +
It is not now difficult to see that the series
lm(X)
converges for
1.
It is possible to use the results (4.15) to obtain a simpler bound by using
further inequalities.
Mn+l
p
n+l
M
p
[v + /3 + 1]
(. + 1]
N
{(( + /3 + 1) + / + I}
=1
=0
p
n+l
( + 1)( +
Un+l
M
n
n
""
n+l
(n
1) (n)
( +
2
M
+ 1)
n
(4.17a)
n
+ 1) + (n + 1)( +
> n( + / + I) + (n + 1)(,8 + B + 1) +
+ 1) +
’
LL. G. CHAMBERS
34
n(= +
+ B + 2) + (6 +
26
n( + } + 1) + ( +
Vn+
(4. 17b)
+ I)
+
(4.17c)
+ S + 1)
Thus
l#m+l(X)l
p M
<
A+;+8+I x=++S+l gn(X)
6 +
(4.18a)
where
{pk+2++2 x++l //{(z
gn {x)
+ 1) }n
+
(4.18b)
nl
Thus
PM
[=1 lm(x)l
6 +
m
A++8+I x=+++I
exp {pAa+2++2
(4.19)
xa++(++l)}
It may be
and from this a bound can be obtained using the expression (4.10d).
Is zero and there Is no free term in the integral
equation (1.1) it is possible if the equation is non-llnear to rewrite the
If a relation of form A
equation in the form (3.1) and proceed as indicated.
holds however, the only solution possible, when the free term is zero, is the zero
noted that,
even if
f(x)
solution.
THE SPECIAL CASE OF A LINEAR EQUATION
In this case
K(x,y) #{y)
g{x,y,#{y}}
5.
(5. I)
and equatlon (1.1} ls of the form
Ax
(x)
f(x) +
/
(5.2a)
K(x,y) #(y) dy
0
The alternatlve form corresponding to equatlon (3.1) Is
Ax
#(x)
k(x) +
f(x}
I0
k(x) + G{x}
(5.2b)
f(y)} dy
K{x,y} {#(y)
(5.2c)
say
where
Ax
[0
k(x)
If
A
(5.2d)
K(x,y)f(y)dy
A holds, that is
and condition
IK(x,y}l
and evldently if
the solution will be unique,
will be the zero solution.
(5. zf)
P(x} Q(y}
f(x}
is zero,
the solution
Furthermore the whole of the theory of section 3 and
section 4 remains valid.
The relation {3.7a}, which is a bounding inequality for
holds,
and,
by analogy,
it follows from equation
l#(x)
f(x)
still
(5.2a}, on taking modull and
applying the Gronwall inequality that
l(x)l
If(x)l
+ P(x)
[
If(Y)l
0
The Iteratlve solution is given by
Q(y}
P(z) Q(z) dz
dy
(5.3)
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
35
x
fCx} + |
0
@n+l (x)
K{x,y)
@n{y}
n > 0
dy
(5.4a}
f(x)
#o(X)
(5.4b)
amd it can be shown that
bn(X)
is of the form
Z ,[_m
n
f(x) +
#n(X)
m=l
x
Km(x,y)
0
f(y) dy
[5.4c]
and K
may be obtained by substituting the relation
m
m
(5.4c} into the Right Hand Side of equation {5.4a}.
A
The values of
Let
@(x}
r
f(x) +
K(x,z)
m
f(z} +
0
m=l
Ax
f(x)+
0
5.5a
dz
AA x
n
Z [
K(x,y) f(y)dy +
Km(z,y
0
m
f(y)
0
m=l
K(x,z)
YA-I
m
Km(z,y)
dz
dy
(5. Sb)
The
reversal
of
conditions on
K
the
order
and
of
is
integration
in
fact
valid
for
fairly
wide
It can be seen that the Right Hand Side of the
f
equation {5.5b} can be rewritten as
_AmX
n+
Z
f(x) +
0
m=l
Km(x,y)
f(y)dy
(5.5c)
provided that
A
and the
K
m
m
A
m
(5.6a)
are defined by the iterative sequence
Ax
Km+l(x,y)
]
K(x,z) K (z y) dz
yA
m
-I
m >
(5.6b)
m
and
Kl(X,y)
K(x,y)
(5.6c)
and thus the solution of equation (5.1), if the appropriate conditions hold is
Ax
f(x) +
Zf
m=l
0
Km(x,y}
f(y) dy
(5.6d)
The error in stopping at the nth term is given again by the relations (4.5)
and (4.9).
Consider now how the theory of sections 3 and 4 can be applied when
and
f(x)
obey the fairly loose condition of boundedness.
K(x,y)
36
LL. G. CHAMBERS
IK(x,y)
< p
(5.7a)
If(x)l
(S. Tb)
It follows immediately that
x
Ik(x)
K(x,y) f(y) dy
(5.7c)
pkx
0
and the
C
defined in (4.4b) is given by
C <
p,c
+ p
e
p (Ac-y)
pAc e
dy
xpc
(5.7d)
0
G(x)
as defined by equation (5.2c) now obeys the inequality
Ax
IGCxI
< p
I
e
p(Ax-y)
[epx- 1]
dy
(5.7e)
0
Thus, the bound given by equation (5.2b) becomes
l(x)
f(x)l
< pkx +
[epkx I]
(5.8a)
and the bound given by the inequality (5.3) becomes
I (x l
l (x)l
}
+
The following results follow for the various quantities defined in equations
(1.4) and (1.6)
o:
0
(5.9a)
3
0
(5, 9b)
’I’
(5.9c)
I
(5.9d)
p
M
(5.9e)
The relevant results from equations (4.9) become
cn
CpAe
n
p
t
s
(5.10a)
2
(5. lOb)
n
n
I)
n(n
n
Apc
n-1
+ n
n
2
+ n
2
(5.10c)
giving
CAeAPC
n+l
(5. lOd)
< p
n!
n
and the error involved in stopping at the nth member of the sequence in the
C
iterative solution is given by
Xn(X)
g
P I!AeApc
n!
A [n2+n] C(px)
The bound given by (4.10) can also be obtained.
n
Equation (4. lOc) gives
(5. fOe)
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
37
g(x,y,f(y)) dy
K(x,y) f(y) dy
(5.11a)
(I + pax)
K
and equation (4.19) gives
giving a further bound
l(x)
p2A3x2 e pA3x
p + pAx +
(5. Iic)
MULTIDIMENSIONAL EQUATION
6.
The theory outlined above may easily be extended to multidimensional systems
of equatlons.
Consider the
n
dimensional system of equations
Ax
(x) + |
Cx)
K(x,Y,(Y))
(6.1)
dy
0
were
(I
(x)
K
@n
satisfies the conditions
llg(x,Y,l)
g(x,Y,2)ll
lla(x,y,a)ll
K P(x)
R(x)
211
Q(y)ll1
(6.2a)
S(y)IIII
(6.2b)
and
MArxa
(6.2c)
g(x, y,(y)) dy
(6.2d)
llk_(x)ll
where
x
|
(x)
0
II II
denotes an appropriate norm.
In exactly the same way as previously it follows that
ll(x)
(x}ll
ll(x)ll
0
+ P(x)
llk(y)ll Q(y)
P(z) Q(z) dz
exp
dy
Y
(6.3)
which corresponds to (3.7a) and
ll(x)ll
ll(x)ll
ll(y)ll S(y)
+ R(x)
exp
R(z) S(z) dz
dy
(6.4)
0
which corresponds to (3.9b).
An iterative sequence function vector may be generated
Ax
n+l(x)
f(x) +
0
g(x,Y,n(y)
dy
n 2 0
(6.5a)
LL. G. CHAMBERS
38
0(x)
Cx)
(6.5b)
The error in stopping at the nth iteration
Zn(X)
IIk_(x)ll replaces
II(x)
is defined as
n(x}ll
(4.4) to (4.9) are obtained,
and identical formulae to those of
save
that
Ik(x)l
Similarly
n
IIn+l(x)ll I1 l(x)ll
IIm(x)ll
+
m=l
(6.6a)
where
m
(x)
m+l(x)
m(X)
(6.6b)
[ IIm(x)ll
(6.6C)
giving
II(x)ll
a bound
P(x)
or
the
Ill(x)
k_(x)ll
+
+
m=l
infinite sum being given by the expression
(4.19),
when
p
For a iinear system, the set of equations assume the form
Ax
(x)
f(x) +
f
(6.7)
l(x,y) (y) dy
0
K(x,y)
where
is a matrix.
One bound is given by
II{(x)
(x)ll < IIk(x}ll
(6.8a)
+ S(x)
where
Ax
and
G(x)
f(y)
is defined by {5.2c),
being replaced by
llf(y)ll
and another
bound is given by
IIt(x)ll
<
II(x)ll
(6.8c1
+ S(x)
The solution sequence becomes
Ax
n+l(x)
f(x) + |
K(x,y)
0
{o(X)
{n(y)
dy
n > 0
(6.9a)
(x)
(6.9b)
p
(6. lOa)
and if
IIK(x,y}ll
II(x}ll
(6. lOb)
the results of (5.7) hold, giving
II(x) -(x)ll
II(x)ll
II(x}ll
pAx + {e pax- I}
II(x)ll +
p +
px
+
pax
{e
I}
p2A3x2epA3x
(6.11a)
(6.11b)
(6. llc)
(6.11b) gives the tighter bound for
It can easily be shown that for small x
(6.11c)
bound.
the
but
tighter
x
for
gives
large
ll(x)ll
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
7.
APPLICATION TO EXAMPLES
A)
Consider now the generallsed pantograph equation
8’ (x)
a 8(Ax} + b 8(x)
39
(7. la
This equation has been discussed extensively [2],
[4], [5].
Let
e(x)
e
bx
(7. Ib)
#(x)
Equation (7.1a) then assumes the form
a exp[b(A
#’ (x)
e(O)
and if
#(0)
(7. Ic)
I)] #(Ax}
this assumes the integral equation form
#(x)
+ ak
Ax
-1
(7. ld)
exp (b’y) (y) dy
0
where
b
b"
b(A
c + i
b"
1)
(7.1e)
c" + i"
(7. lf)
The alternative form for (7.1d) becomes
k(x) + aA -1
(x)
x
exp (b’y)[#(y)
(7.1g)
I] dy
0
where
Ax
aA_l
k(x)
(7.1h)
exp (b’y) dy
0
(7.1g)
Taking moduli, equations (7.1d)
ICx)l
ICx)
Ik(x)l
11
lalA_l
+
Ik(x)l
;Xx
I1 x+
xyx
lalX_l
(7.2c) that
IkCx)l
lal
1
Ilx
/
(Xc)
11
I#(Y)
[exp (c’Ax
-lal
and the inequality (7.2b) becomes
ICx)
-
exp (c’y)
C7.2a)
dy
(7.2b)
dy
1)1
(7.2c)
it can easily be seen from the inequality
zero,
c
(cmy)Icy)l
exp
exp (c’y) dy
0
In the special case of
(7.1h) take the respective forms
lal
C7.2d)
x
x
11
I(Y)
0
(7.2e)
dy
and application of the Gronwald-Bellman-Reid inequality gives
I,Cx)
11
lal x
+
I1 xcl
lalxCl
Clearly,
because of linearity,
lalx_l
[Xx laly
[exp
0
x) +
x) +
;
lalx
(x
y)l dy
Cx
y) dy
Ax
0
I1
exp
XCe lalx
the results for
11
1).
#(0)
(7.2f)
arbitrary can easily be
-
LL. G. CHAMBERS
40
obtained.
The bound given by (3. lOa) becomes, on uslng the inequality (7.2a),
I,cxl
I1
exp
or in the special case of
c
zero
Using the fact that
O(x)
(7.2c)
Ilco’
zl
(7.3c)
can easily be obtained.
{3.7a) becomes,
The bound given by
Icx
(7.3b)
jx I,Cxl
Iocxl
a bound for
(7.3a)
exp (c’z) dz
-
-z txp co.xx
lalCxc,)
0
In the special case of
inequalities
on using the
(7.2b)
and
z
l]exp(c’y)exp
[exp(c’Ay)
-
Xx
lal x-
exp(c’z)dz dy
(7.4a)
zero, the formula (3.7b) may be used, giving
c
Ic 1 [ell ,).
Ax
IlxCl -/-/ I1 exll
0
IlxC
/
<elal
c.
C-I
>.
C.o
Slightly more complicated formulae will follow for the corresponding bound
for
O(x}
B}
Consider now the many dimensional
O
is an
matrices of order
A
dimensional vector and
n
equation
(7.5a)
AOClx) + Be(x)
O’(x)
where
8eneralised pantograph
and
B
are constant complex square
An analytical discussion of this has been given in [6]
n
Bounds associated with this equation may be obtained by extensions of the
methods discussed previously.
Suppose
that,
of
first
all
transformation has been made so that
suitable
a
B
is diagonal.
(possibly
linear
If
B
complex)
is degenerate,
all
that happens is that some of the diagonal terms will be zero.
Then equation (7. Sa) may be rewritten as
n
O’(x)
r
a
s=l
rs
8 (Ax) + b
s
r
8 (x)
r
r
n
(7.5b)
with an obvious notation.
Let
8 (x)
r
exp (b x)
r
(7.5c)
@r{X)
The set of equations (7.5b} then assumes the form
n
exp
(brX} #r(X}
2 mrs exp
s=
(bsAX) sCAX)
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
41
or
n
r(X)
ars exp lrsX
s=!
(7.5d)
s(kX)
where
--FS
=b
(7.5e)
-b/A
F
S
If
Cr(O)
c
(7.5f)
r
equatlon (7.5d) takes the form
n
Cr (x) cr
I
[ ars A -I
s=l
+
Ax
(7.5g)
(,rsY)#s(y)dy
exp
0
The alternative form for equation (7.5g) is
n
#r(X)
kr(X)
Cr
[ ars A -I
s=l
+
x
exp
(rsY) {#s(y)
c
s}
dy
(7.5h)
where
n
I
kr(x}
s=
;
Ax
-I
ars
-
Let
rs
max Re
r,s
Then equation (7.5g) becomes
n
Icl
I,Cx)l
/
7. lasl
s=l
Let
max
la rs
Then the Inequality (7.6b) becomes
Ic r
I(x)l
+ a
exp
0
(rsY)c s
(7.6a)
1
Ax
exp
0
a
c.)I%C)1
d.
s=l
0
(7.6b)
(7.6c)
r
exp (y)
r
(7.5i)
dy
I,s(Y)l
dy.
(7.6d)
Let
n
n
n
[
r=l
a
r
a
[ Icl
r=l
[ I,cx)l
r=l
c
,Cx).
(7.6e)
Then it follows from the inequality (7.6d) that
(x)
c + aA
-I
tax
exp (BY) (Y) dy
(7.6f)
0
It follows immediately that in the same way as previously
(x) <
aA
c exp
-I exp
(z) dz
(7.6g)
A second bound follows from equation (7.5h).
Let
r(X)
c
r
@r(X)
(7.7a)
42
LL. G. CHAMBERS
Then equation (7.5h) assumes the form
Ax
n
r(x)
=k
r{x)
+
0
[ ars A-1
s=1
x
x
In the same way as before
A-
exp
(7.7b)
(,rsY),s(y)sy.
n
(7.7c)
s--1
With an obvious notation, it follows that
-I
(X) K k(x) + aA
Thus,
us
the nequalty (3.6),
Ax
k(x) +
(x)
(7.7d)
exp (BY) (Y) dy
follows that
-1
k(y) aA
exp (y)
aA
exp
-1
exp (z) dz
dy
(7.7e)
0
and, if
is differentiable, the relation (7.7e) can be written as
k(x)
0
O(x)
Clearly, if
0
k’(y)
ak
exp
-1
exp (z) dz
dy
(7.7f)
these formulae simplify.
Alternative bounds, based on the results of section 6 may also be obtained.
Let
where
exp {Bx}
(7.8a)
(x)
exp {Bx}
(x)
is interpreted as
= BSx
+
being the unit matrix of order
s
(7.8b)
)
n
It ls not difficult to see that equatlon (7.5a) assumes the form
exp {B(;t- l)x}
’(x)
(7.8c)
(Xx)
This can be rewritten, using the initial condition as
kx
(x)
(0)
AA-
+
exp (B’y) (y) dy
(7.8d)
0
where
B"
X
B(1
-1
(7.8e)
It follows from (6.8) that
Ax
k(x)
I0
AA -I exp (Bmy) (0) dy
x
AA-
B’Sys
Z0
B’S(Ax)S+1
s=
AA-1
Z
s=0
Thus
’s! (0)
(s + 1)!
(01
(7.8f)
dy
(7.8g)
43
PROPERTIES OF A FUNCTIONAL INTEGRAL EQUATION
IIk(x)ll
If
IIAII
A-1
s=O IIB’llS(Ax)S+l(s
-I1^11
+ 1)!
-
x< c
IIAII
IIk(x)ll <
IIB’I1-1
IItll(o)ll
[expllB’llx>
(7.8h)
exp {lIB IIc}
IIt(o)ll
(7.)
dy
Thus the constants for the inequality (6.2c) are given by
IIAII ),-
{llBllc II(0)II
exp
(7.8j)
8
"
Comparing equation (7.8d) and equation (6.7) it can be seen that
(7.9a)
(0)
f[x)=
and
(7.9b]
exp (B y]
A A
K[x,y)
Now
fiR A -I exp
exp {lIB fly}
y)ll < llAll A
(B
(7.9c)
and as
(7.9d)
y K Ax K Ac
it follows that
IIAII
IIK(x,y)ll
Thus, the quantities
p
X
-1
exp
{IIB’II
(7.9e)
Xc}
defined in (6.10) are given by
and
IIAII
p
X
-1
(IIB’II
exp
(7.9f)
and
IIC0}ll
Now the relations (6.11) will hold for
x
(7.9g)
In particular,
x K c
they hold for
c
The inequality (6. lla) becomes
l(c)
[epAc 1]
fCc)ll < pAc +
(7.10a)
The inequality (6.11b) becomes
IICc)ll
<
II_fCc)ll
+
[epXc 11
(7. lOb)
and the inequality (6.11c) becomes
IIt[c)ll
where
p
is
K p + pAc +
in fact a function of
p2A3c2e pA3c
c
defined by (7.9f).
(7.10c)
c
arbitrary and so the inequalities (7.10) give bounds for all positive
8.
however
is
c
DISCUSSION
Because of the way in which the bounds discussed in this paper have been
derived, namely by means of a generalisation of Gronwall’s inequality, they
involve exponentials with positive coefficients,
associated with an
increasing
LL. G. CHAMBERS
44
divergence from the initial values of the dependent variable as the independent
variable increases.
Consequently, they would not be suitable, except near the
initial value of the independent variable for discussing problems such as that
defined by
dy
d--
y(O)
y(x)
(8.1a)
which is equivalent to the integral equation
A -1
lAX
(S. Ib)
y(u) du
0
Generally, where solutions are asymptotically stable, and converge to some limit
y(x)
for large
x
the bounds discussed here will become irrelevant for large enough
x
This would be equally true of multidimensional
equations for which the
solutions are asymptotically stable.
If, however, the equations are such that solutions are unstable
as would,
B
matrices of
for example, be the case when all the elements of the
(7.5a) are positive
A
and
the bounds here will always be relevant.
that sometimes one bound is better,
sometimes another.
seen, without much difficulty that if
c
is near zero,
It may be noted
For example,
it can be
the bound given by the
inequality (7.10b) is tighter than that given by the inequality (7.10c) whereas if
c
is large, the reverse situation holds.
ACKNOWLEDGEMENT.
am grateful
to Professor Joel Rogers for comments on a
previous version of this paper.
KEFENENCES
1.
e.g. HALE, Jack, Theory of Functional Differential Equations, SpringerVerlag, Berlin, 1977.
2.
KATO,
y’(x)
3.
e.g. RAO, M Rama Mohana,
London, 1981, 40.
4.
CARR,
y’(x)
5.
as 5. in references on p174 of Carr/Dyson paper.
CARR, Jack and DYSON, Janet, The Matrix Functional Differential Equation
Ay(Ax) + By(x), Proc Roy Soc Ed 75A (1975/6), 5-22.
y’(x)
6.
J B, The functional differential
and McLEOD,
ay(Ax) + by(x), Bull Amer Math Soc 77 (1971), 891-937.
Tosio
Ordinary Differential Equations,
Edward
Differential
Functional
The
and
Janet,
DYSON,
ay(Ax) + by(x), Proc Roy Soc Ed 74A (1974/5), 165-174.
Jack
equation
Arnold,
Equation